In `DeltaABC,D` and E are points on the sides AB and AC respectively ,such that DE||BC. If AD=2.5cm,BD=3cm and AE =3.75 cm , then the value of AC.

Answer

To solve the problem step by step, we will use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- In triangle \( ABC \), points \( D \) and \( E \) are on sides \( AB \) and \( AC \) respectively.
- Given lengths:
- \( AD = 2.5 \, \text{cm} \)
- \( BD = 3 \, \text{cm} \)
- \( AE = 3.75 \, \text{cm} \)

2. **Calculate the Length of \( AB \):**
- Since \( AB = AD + BD \):
\[ AB = AD + BD = 2.5 \, \text{cm} + 3 \, \text{cm} = 5.5 \, \text{cm} \]

3. **Use the Property of Parallel Lines:**
- Since \( DE \parallel BC \), triangles \( ADE \) and \( ABC \) are similar by the AA (Angle-Angle) similarity criterion. Therefore, the ratios of corresponding sides are equal:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]

4. **Substitute the Known Values:**
- Substitute \( AD \), \( AB \), and \( AE \) into the proportion:
\[ \frac{2.5}{5.5} = \frac{3.75}{AC} \]

5. **Cross-Multiply to Solve for \( AC \):**
- Cross-multiplying gives:
\[ 2.5 \cdot AC = 3.75 \cdot 5.5 \]

6. **Calculate \( 3.75 \cdot 5.5 \):**
- First, calculate \( 3.75 \cdot 5.5 \):
\[ 3.75 \cdot 5.5 = 20.625 \]

7. **Solve for \( AC \):**
- Now, substitute back into the equation:
\[ 2.5 \cdot AC = 20.625 \]
- Divide both sides by \( 2.5 \):
\[ AC = \frac{20.625}{2.5} = 8.25 \, \text{cm} \]

### Final Answer:
The length of \( AC \) is \( 8.25 \, \text{cm} \).

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